Scalar (mathematics)

A scalar is a mathematical quantity that is characterized solely by the specification of a numerical value (in physics with unit, if necessary).

In the mathematical subfield of linear algebra, scalar denotes an element of the basic body of a vector space, usually a real number. In contrast, the elements of a vector space are called vectors. Accordingly, the basic body is also called the scalar body. Multiplying a vector by a scalar λ is called scalar multiplication or scaling. The resulting vector λ is called a scalar multiple of .

In contrast to scalar multiplication, the scalar product is a linkage that assigns a scalar as a value to two vectors.

The term scalar goes back to the Latin word scala (ladder) in the sense of a uniform division (see scale).

Scalars in physics

In physics, scalars are used to describe physical quantities that are independent of direction. Examples of scalar physical quantities are the mass of a body, its temperature, its energy and also its distance from another body (as the magnitude of the difference of the position vectors). In other words, a scalar physical quantity does not change with changes in location or orientation. If, on the other hand, a direction is required for the complete description of the quantity, as in the case of the force or the velocity, a vector is used, and in the case of dependence on several directions, a tensor (more precisely: tensor of the 2nd level or even higher).

The velocity of a particle has the direction in which the particle is moving. Since the direction changes with rotations, the velocity is not a scalar, but a vector. But the magnitude of the velocity does not change with rotations and is a scalar.

Whether a quantity is a scalar depends on the transformation group under consideration. Thus energy is a scalar with respect to rotations, but in relativity it is a component of a four-vector.

A subgroup of the scalars are the pseudoscalars, which reverse the sign under a plane reflection.

Extensions and delimitation of similar terms

• Quadratic matrices, which (taken as a linear mapping of a vector space onto itself) correspond to a multiplication of each vector by a fixed scalar λ to have the property scalar. They are diagonal matrices whose entries on the diagonal are all equal to λ .

• Also in a module over a ring the multiplication of a module element with an element of the base ring is called scalar multiplication. However, the term scalar for the elements of the base ring is only partially used in this case.